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Pré-Publication, Document De Travail Année : 2023

A pseudodifferential calculus for maximally hypoelliptic operators and the Helffer-Nourrigat conjecture

Résumé

We extend the classical regularity theorem of elliptic operators to maximally hypoelliptic differential operators. More precisely, given vector fields X1,. .. , Xm on a smooth manifold which satisfy Hörmander's bracket generating condition, we define a principal symbol for any linear differential operator. Our symbol takes into account the vector fields Xi and their commutators. We show that for an arbitrary differential operator, its principal symbol is invertible if and only if the operator is maximally hypoelliptic. This answers affirmatively a conjecture due to Helffer and Nourrigat. Our result is proven in a more general setting, where we allow each one of the vector fields X1,. .. , Xm to have an arbitrary weight. In particular, our theorem generalizes Hörmander's sum of squares theorem to higher order polynomials. Contents A Proofs of Theorems 2.9 and 2.4 A.
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Dates et versions

hal-03987186 , version 1 (14-02-2023)

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  • HAL Id : hal-03987186 , version 1

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Androulidakis Iakovos, Mohsen Omar, Robert Yuncken. A pseudodifferential calculus for maximally hypoelliptic operators and the Helffer-Nourrigat conjecture. 2023. ⟨hal-03987186⟩
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